Fuzzy data envelopment analysis based upon fuzzy arithmetic with an application to performance assessment of manufacturing enterprises q Ying-Ming Wang a, * , Ying Luo b , Liang Liang c a School of Economics & Management, Tongji University, Shanghai 200092, PR China b School of Management, Xiamen University, Xiamen 361005, PR China c School of Management, University of Science and Technology of China, Hefei 230026, PR China article info Keywords: Data envelopment analysis Fuzzy input and fuzzy output data Fuzzy efficiency Fuzzy ranking approach Fuzzy arithmetic abstract Data envelopment analysis (DEA) requires input and output data to be precisely known. This is not always the case in real applications. This paper proposes two new fuzzy DEA models constructed from the perspective of fuzzy arithmetic to deal with fuzziness in input and output data in DEA. The new fuzzy DEA models are formulated as linear programming models and can be solved to determine fuzzy efficien- cies of a group of decision-making units (DMUs). An analytical fuzzy ranking approach is developed to compare and rank the fuzzy efficiencies of the DMUs. The proposed fuzzy DEA models and ranking approach are applied to evaluate the performances of eight manufacturing enterprises in China. Ó 2008 Elsevier Ltd. All rights reserved. 1. Introduction Traditional data envelopment analysis (DEA) models such as CCR and BBC models (Banker, Charnes, & Cooper, 1984; Charnes, Cooper, & Rhodes, 1978) require crisp input and output data. In real world situations, however, crisp input and output data may not always be available, especially when a set of decision-making units (DMUs) contains missing data, judgment data, or predictive data. Generally speaking, uncertain information or imprecise data can be characterized by fuzzy numbers, which include interval numbers as a special case. Therefore, how to evaluate the efficien- cies of a group of DMUs with fuzzy input and output data is a prob- lem worthy of study. Several attempts have been made to deal with fuzzy input and output data in DEA. For example, Sengupta (1992) proposed a fuz- zy mathematical programming approach which incorporated fuzz- iness into a DEA model by defining tolerance levels on both objective function and constraint violations. Triantis and Girod (1998) suggested a mathematical programming approach through transforming fuzzy input and output data into crisp data using membership function values. Efficiency scores were computed for different values of membership functions and then averaged. Guo and Tanaka (2001) proposed a fuzzy CCR model in which fuzzy constraints including fuzzy equalities and fuzzy inequalities were converted into crisp constraints by predefining a possibility level and using the comparison rule for fuzzy numbers. León, Liern, Ruiz, and Sirvent (2003) suggested a fuzzy BCC model based on the same idea. Lertworasirikul, Fang, Joines, and Nuttle (2003a) proposed a possibility approach which deals with uncertainties in fuzzy objec- tives and fuzzy constraints through the use of possibility measures. It transforms a fuzzy DEA model into a well-defined possibility DEA model. In the special case that fuzzy data are trapezoidal fuzzy numbers, the possibility DEA model becomes a linear program- ming model. They (Lertworasirikul, Fang, Joines, & Nuttle, 2003b) also proposed a credibility approach as an alternative way to solve the fuzzy DEA model. The credibility approach transforms the fuz- zy DEA model into a well-defined credibility programming model, in which fuzzy variables were replaced by expected credits in terms of credibility measures. The expected credits of fuzzy vari- ables were derived by using credibility measures, which are the averages of possibility and necessity measures. The possibility and credibility approaches were further extended to fuzzy BCC model in Lertworasirikul, Fang, Nuttle, and Joines (2003) by the same authors. Wu, Yang, and Liang (2006) applied the possibility DEA model for efficiency analysis of cross-region bank branches in Canada. Garcia, Schirru, and Melo (2005) utilized the possibility DEA model for failure mode and effects analysis (FMEA) and pre- sented a fuzzy DEA approach to determining ranking indices among failure modes. Kao and Liu (2000b, 2003, 2005) transformed fuzzy input and output data into intervals by using a-level sets and Zadeh’s exten- sion principle, and built a family of crisp DEA models for the inter- vals. Based upon their crisp DEA models for a-level sets, Liu (2008) 0957-4174/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2008.06.102 q The work described in this paper was supported by the National Natural Science Foundation of China (NSFC) under the Grants No.70771027 and 70525001. * Corresponding author. Tel.: +86 591 87893307. E-mail address: msymwang@hotmail.com (Y.-M. Wang). Expert Systems with Applications 36 (2009) 5205–5211 Contents lists available at ScienceDirect Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa